Answered by thomas via Travellr.com
I stayed at Rich Lanna House recently. It's a newish guesthouse on the edge of the old city wall.
The quality of the rooms was excellent for the price. Location was convenient and place has a small pool and reasonable buffet breakfast.
I think their smallest rooms will be around $40. (Perhaps you can negotiate depending how many nights you are staying).
Hope that helps about 6 years ago
Oops - sorry. Mis-read your message and answered for Chiang Mai. Doh.
Last time visiting Chiang Rai I stayed at Dusit Thani. Advertised rates are higher but if you catch one of their promotions it should be around your $40 range. (Wotif mystery deal or latestays perhaps?).
It's walkable to the city centre just about, plus they have a shuttle. Rooms a little dated but the facilities are good.
It's on it's own island in the river which is kind of cool. about 6 years ago